If you are using Easel to control your X-Carve then each time you wish to use Easel you must have Easel running (internet connection) and your X-Carve connected to the computer that is running Easel (USB connection).

If you are re-carving a previous Easel project, you just need to reload that saved project once everything is connected.

So the short answer to your question is yes you do need to reconnect, nothing is stored in the browser for your project.

When you load Easel, it loads all the modules it needs to generate tool paths into the browser. The communication with the machine happens between the Easel webapp and the Easel driver that you installed onto your computer. The internet connection is only needed to save changes to your project.

The Easel System Requirements state the following…
“Internet access is required to use Easel. You need an Internet connection when using Easel to design projects and send files to your machine. We understand that many users would like to download Easel directly to their computer, but unfortunately, Easel is too complex to accommodate a local program.”

Where can I find the support article you are referring to here?
I am still not clear on when and where an internet connection, a computer, AND the X-Carve all need to be present to design and run a project.
Can I design a project on an internet connected computer without the X-Carve being present/connected?
Once designed and auto saved on Easel, can I run the project in an area without internet connection where my X-Carve is located?
Can I save a GCode download of a design to a file on my computer and run that file from my computer when connected to X-Carve in an area where there is no internet connection?
Thanks,

Easel is very user friendly. but not necessary to run a gcode file through the X-Controller. One feature I like about Easel is the beginning steps to setuping up your project. Other then that, pretty much the same once you start carving.